Page 144 - Wind Energy Handbook

118 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

The momentum theory, as developed, applies only to the whole rotor disc where
the flow induction factor a is the average value for the disc. However, it may be
argued that it is better to apply the momentum equations to an annular ring, as in
the non-yawed case, to determine a distribution of the flow induction factors
varying with radius, reflecting radial variation of circulation. Certainly, for the
angular momentum case the tangential flow factor a9 will not vary with azimuth
position because it is generated by the root vortex and although, in fact, the axial
flow factor a does vary with azimuth angle it is consistent to use an annular average
for this factor as well.
To find an average for an annular ring the elemental values of force and moment
must be integrated around the ring.
For the axial momentum case, taking the vortex method as an example,

ð 2ð
1 2 ÷ 2 ÷
rU 4af cos ª þ tan sin ª af sec rär dł
1
2 2 2
0
ð 2ð
2
¼ 1 rW ó r C x rär dł
0 2

Therefore

ð
÷ ÷ 2ð W 2
8ðaf cos ª þ tan sin ª af sec 2 ¼ ó r C x dł (3:137)
2 2 0 U 2 1

The resultant velocity W and the normal force coefficient C x are functions of ł. And
for the angular momentum case

ð 2ð
1 2
2
2
2
2
rU ºì4a9 f(cos ª af)(cos ł þ cos ÷ sin ł)r är dł
2 1
0
ð 2ð
2
2
¼ 1 rW ó r (cos ł sin ÷C x þ cos ÷C y )r är dł
0 2
which reduces to
ð 2ð 2
2 W
4a9 f(cos ª af)ºìð(1 þ cos ÷) ¼ ó r 2 (cos ł sin ÷C x þ cos ÷C y )dł (3:138)
0 U 1

The non-dimensionalized resultant velocity relative to a blade element is given by